A smooth local family of extremals with prescribed endpoint exists when the endpoint momentum map has invertible derivative

By [F1] and [F2], the solution map of Hamilton's equations is smooth in the initial state near $(q_0,p_0^{\ast })$ on a common open solution interval $K$ containing $t_0$ and $t_1$. The local conclusion is propagated along the reference solution as follows. Choose a compact time interval strictly inside its lifetime containing both times in its interior. Local smooth solution neighbourhoods from [F2] cover the compact reference trajectory. Choose around each reference time $u$ a radius $r_u>0$ so small that initial pairs along the reference curve for $\mathrm{\mid }s-u\mathrm{\mid }<2r_u$ lie in its local smooth-solution domain, and that this local map evaluates solutions throughout that larger interval. The smaller intervals $\mathrm{\mid }s-u\mathrm{\mid }<r_u$ cover the compact time segment, so finitely many suffice. A subdivision of mesh less than the minimum of these finitely many radii, with $t_0$ inserted, has each short piece in one larger interval: its first endpoint lies in a smaller interval and every point of that piece differs from it by less than the corresponding radius. Compose the local solution maps, shrinking the initial-state neighbourhood at each of the finitely many joins so its image stays in the next domain. Uniqueness identifies the composition with the initial-value solution; the same construction backwards covers earlier times. This supplies smooth dependence on a common open interval containing the compact segment. Thus in particular the endpoint map $E$ is defined and smooth on a neighbourhood of $(t_1,p_0^{\ast })$, and $F(t,p_0)=(t,E(t,p_0))$ is a smooth map whose derivative at $(t_1,p_0^{\ast })$ is invertible by the hypothesis and [F3].

Since $E(t_0,p_0)\equiv q_0$, its momentum derivative at $t_0$ is zero; invertibility implies $t_1\mathrm{\ne }{t}_0$, so choose $J_1$ to exclude $t_0$. By the inverse function theorem with higher regularity [F3], there are open neighbourhoods $J_1\ni t_1$, $Q_1\ni q_1$ and $P_1\ni p_0^{\ast }$ and a $C^{\mathrm{\infty }}$ map $p_0:J_1\times Q_1\to P_1$ with $p_0(t_1,q_1)=p_0^{\ast }$ and $F(t,p_0(t,q))=(t,q)$, i.e. $E(t,p_0(t,q))=q$, for all $(t,q)\in J_1\times Q_1$.

Define $X(t,q,s)$ as the $q$-component at time $s$ of the Hamiltonian solution with data $(q_0,p_0(t,q))$ at $t_0$. By [F2] the local solution map is smooth in the initial state and in the evaluation time, so, after shrinking $J_1$ and $Q_1$ so that $p_0(t,q)$ remains in the region of smooth dependence and so that all these solutions exist on a common open interval $K\subseteq J$ containing $t_0$ and $J_1$, the map $X$ is $C^{\mathrm{\infty }}$ on $J_1\times Q_1\times K$.

Properties of $X$: $X(t,q,t_0)=q_0$, because every solution starts at $q_0$ at time $t_0$; and $X(t,q,t)=E(t,p_0(t,q))=q$ by step 2.1. For each $(t,q)$, the curve $s\mapsto (X(s),p(s))$ solves Hamilton's equations with $p$ the momentum component of the same solution, so by [F4] $s\mapsto X(t,q,s)$ solves the Euler–Lagrange equation of $\mathcal{L}$ with velocity ${\mathrm{\partial }}_{s}X$.

Steps 3.1 and 4.1 show that $X$ satisfies the hypotheses of The endpoint action function of a smooth family of extremals has differential $dW=pdq-Hdt$ on $J_1\times Q_1$; applying that theorem gives that $W(t,q):={\int }_{t_0}^t\mathcal{L}(s,X,{\mathrm{\partial }}_{s}X)ds$ is $C^1$ with $W_q(t,q)=p(t,q)$ and $W_t(t,q)=-H(t,q,p(t,q))$, where $p(t,q)={\mathcal{L}}_v(t,q,{\mathrm{\partial }}_{s}X(t,q,t))$.